![](https://blog.fridaymath.com/wp-content/uploads/2022/11/rmm-2-altitudeRightBisectors.png)
Right now we’ll use this problem to obtain another characterization of right triangles, in addition to the ones here.
Stretched image
To start with, here’s an enlarged version of the above image:
![](https://blog.fridaymath.com/wp-content/uploads/2022/11/rmm-2-altitudeRightBisectors.png)
In any triangle, the ratio of the length of an internal altitude to the “length” of a right bisector is always sandwiched between and
.
Strict inequalities
In general, the ratio under consideration lies strictly between and
.
![Rendered by QuickLaTeX.com ABC](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-b5002f05fd90c80f81a9e0e9c845e02d_l3.png)
![Rendered by QuickLaTeX.com F](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-45c58744723006803a0846d88fc288d1_l3.png)
![Rendered by QuickLaTeX.com A](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-3404ae68a19d10407c93ea3824613e4e_l3.png)
![Rendered by QuickLaTeX.com M](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-3d48ed37dcc50d1a0b3d189e3ca50a57_l3.png)
![Rendered by QuickLaTeX.com BC](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-c4f3ce61859dcd00e52fbf771b86cb55_l3.png)
![Rendered by QuickLaTeX.com BC](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-c4f3ce61859dcd00e52fbf771b86cb55_l3.png)
![Rendered by QuickLaTeX.com AC](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-2509ee19f0ee692f93e2b93071b678f5_l3.png)
![Rendered by QuickLaTeX.com N](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-55a8b451f85dacca12e4dd6869a45cca_l3.png)
![Rendered by QuickLaTeX.com AC> AB](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-4af1479f7e4f4d89e60ff699d6bfe34d_l3.png)
![Rendered by QuickLaTeX.com \frac{AF}{MN}=\frac{a^2+b^2-c^2}{a^2}](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-bd9b01442d736d8c61a5a26df67a5f11_l3.png)
Details here.
![Rendered by QuickLaTeX.com ABC](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-b5002f05fd90c80f81a9e0e9c845e02d_l3.png)
![Rendered by QuickLaTeX.com F](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-45c58744723006803a0846d88fc288d1_l3.png)
![Rendered by QuickLaTeX.com A](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-3404ae68a19d10407c93ea3824613e4e_l3.png)
![Rendered by QuickLaTeX.com M](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-3d48ed37dcc50d1a0b3d189e3ca50a57_l3.png)
![Rendered by QuickLaTeX.com BC](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-c4f3ce61859dcd00e52fbf771b86cb55_l3.png)
![Rendered by QuickLaTeX.com BC](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-c4f3ce61859dcd00e52fbf771b86cb55_l3.png)
![Rendered by QuickLaTeX.com AC](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-2509ee19f0ee692f93e2b93071b678f5_l3.png)
![Rendered by QuickLaTeX.com N](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-55a8b451f85dacca12e4dd6869a45cca_l3.png)
![Rendered by QuickLaTeX.com AC> AB](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-4af1479f7e4f4d89e60ff699d6bfe34d_l3.png)
![Rendered by QuickLaTeX.com \frac{AF}{MN}>1](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-dffffdd443d8836c40aefa6b683ec656_l3.png)
Details here.
![Rendered by QuickLaTeX.com ABC](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-b5002f05fd90c80f81a9e0e9c845e02d_l3.png)
![Rendered by QuickLaTeX.com F](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-45c58744723006803a0846d88fc288d1_l3.png)
![Rendered by QuickLaTeX.com A](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-3404ae68a19d10407c93ea3824613e4e_l3.png)
![Rendered by QuickLaTeX.com M](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-3d48ed37dcc50d1a0b3d189e3ca50a57_l3.png)
![Rendered by QuickLaTeX.com BC](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-c4f3ce61859dcd00e52fbf771b86cb55_l3.png)
![Rendered by QuickLaTeX.com BC](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-c4f3ce61859dcd00e52fbf771b86cb55_l3.png)
![Rendered by QuickLaTeX.com AC](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-2509ee19f0ee692f93e2b93071b678f5_l3.png)
![Rendered by QuickLaTeX.com N](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-55a8b451f85dacca12e4dd6869a45cca_l3.png)
![Rendered by QuickLaTeX.com AC> AB](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-4af1479f7e4f4d89e60ff699d6bfe34d_l3.png)
![Rendered by QuickLaTeX.com \frac{AF}{MN}<2](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-25b52069deccea78319833677ce18940_l3.png)
Details here.
Together we have: .
Special instance
When the ratio under consideration equals .
![Rendered by QuickLaTeX.com ABC](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-b5002f05fd90c80f81a9e0e9c845e02d_l3.png)
![Rendered by QuickLaTeX.com \angle B=90^{\circ}](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-499f86f422f50a5b1e41f07f4a245dd7_l3.png)
![Rendered by QuickLaTeX.com M](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-3d48ed37dcc50d1a0b3d189e3ca50a57_l3.png)
![Rendered by QuickLaTeX.com BC](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-c4f3ce61859dcd00e52fbf771b86cb55_l3.png)
![Rendered by QuickLaTeX.com BC](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-c4f3ce61859dcd00e52fbf771b86cb55_l3.png)
![Rendered by QuickLaTeX.com AC](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-2509ee19f0ee692f93e2b93071b678f5_l3.png)
![Rendered by QuickLaTeX.com N](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-55a8b451f85dacca12e4dd6869a45cca_l3.png)
![Rendered by QuickLaTeX.com AC> AB](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-4af1479f7e4f4d89e60ff699d6bfe34d_l3.png)
![Rendered by QuickLaTeX.com \frac{AB}{MN}=2](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-84b58a7725fb43581101a23c1410424c_l3.png)
Use similar triangles. Or else use the fact that: from example 1. Take
so that
. Simplify to get
.
![Rendered by QuickLaTeX.com ABC](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-b5002f05fd90c80f81a9e0e9c845e02d_l3.png)
![Rendered by QuickLaTeX.com F](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-45c58744723006803a0846d88fc288d1_l3.png)
![Rendered by QuickLaTeX.com A](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-3404ae68a19d10407c93ea3824613e4e_l3.png)
![Rendered by QuickLaTeX.com M](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-3d48ed37dcc50d1a0b3d189e3ca50a57_l3.png)
![Rendered by QuickLaTeX.com BC](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-c4f3ce61859dcd00e52fbf771b86cb55_l3.png)
![Rendered by QuickLaTeX.com BC](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-c4f3ce61859dcd00e52fbf771b86cb55_l3.png)
![Rendered by QuickLaTeX.com AC](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-2509ee19f0ee692f93e2b93071b678f5_l3.png)
![Rendered by QuickLaTeX.com N](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-55a8b451f85dacca12e4dd6869a45cca_l3.png)
![Rendered by QuickLaTeX.com AC> AB](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-4af1479f7e4f4d89e60ff699d6bfe34d_l3.png)
![Rendered by QuickLaTeX.com \frac{AF}{MN}=2](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-ef955d9de36ff8343c3c45cc4c456094_l3.png)
![Rendered by QuickLaTeX.com ABC](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-b5002f05fd90c80f81a9e0e9c845e02d_l3.png)
![Rendered by QuickLaTeX.com \angle B=90^{\circ}](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-499f86f422f50a5b1e41f07f4a245dd7_l3.png)
Details here.
Takeaway
Let be a triangle in which
is the foot of the internal altitude from vertex
,
is the midpoint of side
, and the right bisector of
intersects
(or
, depending on which is greater) at
. Then the following statements are equivalent:
is a right triangle in which
.
Tasks
- (Silver ratio) Suppose that
is a triangle containing an obtuse angle
. Let
be the foot of the altitude from
and let the right bisector of
intersect
at
.
- PROVE that
, where
is the midpoint of
.
- Find a triangle for which
, the silver ratio.
- PROVE that